332 
DRS. J. AND E. HOPKINSON ON" DYNAMO-ELECTRIC MACHINERY. 
to accord with the observations.* Such empirical formulae possess no advantage 
over the graphical method aided by algebraic processes, and tend to mask much that 
is of importance. 
One purpose of the present investigation is to give an approximately complete con¬ 
struction of the characteristic curve of a dynamo of given form from the ordinary laws 
of electro-magnetism and the known properties of iron, and to compare the result of 
such construction with the actual characteristic of the machine. The laws of electro¬ 
magnetism needed are simply (Thomson, papers on ‘Electrostatics and Magnetism 
Maxwell, ‘Electricity and Magnetism,’ vol. 2, pp. 24, 26, and 143), (1) that the line 
integral of magnetic force around any closed curve, whether in iron, in air, or in both, is 
equal to 47mc where c is the current passing through the closed curve, and n is the 
number of times it passes through; (2) the solenoidal condition for magnetic induc¬ 
tion, that is, if the lines of force or of induction be supposed drawn, then the induction 
through any tube of induction is the same for every section. Regarding the iron itself, 
we require to know from experiments on the material in any shape the relation 
between a, the induction, and a, the magnetic force at any point; for convenience 
write a —f~ l (a), or a =/(«). From these premises, without any further assumption, 
it is easy to see that a sufficiently powerful and laborious analysis would be capable of 
deducing the characteristic of any dynamo to any desired degree of accuracy. This 
we do not attempt, as even, if successful, the analysis would not be likely to throw 
any useful light on the practical problem. We shall calculate the characteristic, first 
making certain assumptions to simplify matters. We shall next point out the nature of 
the errors introduced by these assumptions, and make certain small corrections in the 
method to account for these sources of error, merely proving that the amount of these 
corrections is probable or deducing it from a separate experiment, and again compare 
the theoretical and the actual characteristic. 
First approximation .—Assume that by some miracle the tubes of magnetic induc¬ 
tion are entirely confined to the iron excepting that they pass directly across from the 
bored faces of the pole-pieces to the cylindrical face of the armature core. This, we 
shall find, introduces minor sources of error, affecting different parts of the charac¬ 
teristic curve to a material extent. Let I be total induction through the armature, 
A x the area of section of the iron of the armature, l x the mean length of lines of force 
in the armature ; A-, the area of each of the two spaces between core of armature and 
the pole-pieces of the magnets, l 2 the distance between the core and the pole-piece ; 
[* Added Aug. 17.—That Frohlioh’s formula cannot be a thoroughly satisfactory expression of the 
characteristic of a dynamo machine is evident from the consideration that E should simply change its 
sign with c, that is, be an odd function of c. There should be a point of inflexion in the characteristic 
E c . 
curve at the origin. Another empirical formula — = tan 1 ^ is free from this objection, but still fails to 
fully represent the approximation of the curve to a straight line on either side of the origin, and it is 
equally uninstructive with any other purely empirical formula.] 
